The **Rationalize the Denominator Calculator** is used for the process of rationalizing the denominator. The presence of a radical in the denominator makes the calculations difficult so it is best to rationalize the denominator.

Rationalizing the denominator means **removing radicals** from the denominator. The radicals include square root and cube root of a number.

If a value with the **cube root** or **square root** is present in the denominator, applying different methods to remove them is called rationalization.

Multiplying and dividing the fraction with the conjugate of the denominator and further simplifying the expression **rationalizes** the denominator.

This calculator rationalizes the denominator and shows the resultant fraction as the output.

## What Is a Rationalize the Denominator Calculator?

**The Rationalize the Denominator Calculator is an online tool that is used to rationalize the denominator of such a fraction with radicals such as square root and cube root in the denominator.**

There are various methods to remove the radical from the denominator depending upon the **type of radical** present.

If a radical such as $ \sqrt{2} $ is present in denominator, **multiplying** and **dividing** by $ \sqrt{2} $ and simplifying the fraction rationalizes the denominator.

If a radical such as $ 2 + \sqrt{3} $ is present in the denominator, this gives rise to the concept of “**conjugate**”. The conjugate of a radical expression is the additive inverse of the radical in the radical expression.

For example, the conjugate of $ 2 + \sqrt{3} $ is $ 2 \ – \ \sqrt{3} $. Notice that the conjugate is not the **additive inverse** of the whole expression but only of the radical itself in the expression.

## How To Use the Rationalize the Denominator Calculator

The user can use the Rationalize the Denominator Calculator by following the steps given below.

### Step 1

The user must first enter the numerator of the fraction in the input tab of the calculator. It should be entered in the block titled “**Enter Numerator:**” in the calculator’s input window.

The numerator doesn’t need to be free of radicals such as square root, cube root, and fourth root.

For the **default** example, the calculator uses 1 in the numerator of the fraction whose denominator needs to be rationalized.

### Step 2

The user must now enter the denominator in the calculator’s input tab. It should be entered in the block labeled “**Enter Denominator:**” in the input window of the calculator.

The denominator must contain a **radical** which is rationalized by the calculator.

If a radical expression such $ \sqrt{3} $ is **not present** in the denominator, the calculator prompts “Not a valid input; please try again”.

The calculator takes $ 4 \ – \ \sqrt{2} $ in the denominator for the default example. The radical in it is $ \sqrt{2} $.

### Step 3

The user must now press the button “**Rationalize denominator**” for the calculator to process the numerator and denominator.

### Output

The calculator takes the input fraction and outputs the fraction by rationalizing the denominator. The output of the calculator shows the following **two windows**.

#### Input

The Input window shows the input interpretation of the calculator. It shows the entered numerator and denominator in **fraction **form.

For the **default** example, it shows the Input as follows:

\[ Input = \frac{1}{ 4 \ – \ \sqrt{2} } \]

#### Alternate Forms

The calculator **rationalizes the denominator** of the entered fraction and displays the alternate form of the fraction in this window.

It removes the radical expression from the denominator by multiplying and dividing the fraction with its conjugate.

The user can view all the **mathematical steps** by pressing “Need a step-by-step solution to this problem?”

For the **default** example, the conjugate of $ 4 \ – \ \sqrt{2} $ is $ 4 + \sqrt{2} $. Multiplying and dividing the fraction by $ 4 + \sqrt{2} $ gives:

\[ Input = \frac{1}{ 4 \ – \ \sqrt{2} } \left( \frac{ 4 + \sqrt{2} }{ 4 + \sqrt{2} } \right) \]

Using the formula:

**( a + b )(a – b ) = $a^2$ – $b^2$**

And simplifying gives:

\[ Input = \frac{ 4 + \sqrt{2} }{ 4^2 \ – \ {(\sqrt{2})}^2 } \]

\[ Input = \frac{ 4 + \sqrt{2} }{ 16 \ – \ 2 } \]

The calculator shows the **alternate form** as given below:

\[ Alternate \ Form = \frac{1}{14} ( 4 + \sqrt{2} ) \]

## Solved Examples

The following examples are solved through the Rationalize the Denominator Calculator.

### Example 1

Rationalize the denominator of the fraction given below.

\[ \frac{2}{ 3 \ – \ \sqrt{5} } \]

### Solution

The user should first enter the **numerator** and **denominator** in the input window of the calculator. The numerator is 2 and the denominator is $ 3 \ – \ \sqrt{5} $ in the example.

After pressing “**Rationalize denominator**”, the calculator computes the output as follows:

The **Input** window shows the fraction whose denominator needs to be rationalized. It interprets the input as follows:

\[ Input = \frac{2}{ 3 \ – \ \sqrt{5} } \]

The calculator shows the **Alternate form** of the expression after rationalizing the denominator as follows:

\[ Alternate \ Form = \frac{1}{2} ( 3 + \sqrt{5} ) \]

### Example 2

The fraction given below contains a radical:

\[ \frac{4 + \sqrt{3} }{ 4 \ – \ \sqrt{3} } \]

### Solution

The numerator $ 4 + \sqrt{3} $ and denominator $ 4 \ – \ \sqrt{3} $ is entered in the calculator’s input window. After submitting the input, the calculator rationalizes the denominator and shows the output as given below.

The **Input** interpretation shown by the calculator is as follows:

\[ Input = \frac{4 + \sqrt{3} }{ 4 \ – \ \sqrt{3} } \]

The calculator rationalizes the denominator by multiplying and dividing with the conjugate of the denominator that is $ 4 + \sqrt{3} $ and simplifies the fraction.

It displays the **Alternate form** of the fraction as follows:

\[ Alternate \ Form = \frac{1}{13} ( 19 + 8 \sqrt{3} ) \]